Question

which of the following situations does not use the sas similarity criterion?
a. verifying similarity between two mountain slopes
b. proving congruence of two triangles
c. determining the similarity of two bird nests based on size ratios
d. proving that two bridges have proportional supports

in triangle pqr and triangle stu, if ∠p = ∠s and pq/st = pr/su, what can be concluded?
a. triangles pqr and stu are congruent by sas congruence
b. triangles pqr and stu are similar by aaa similarity
c. triangles pqr and stu are not similar
d. triangles pqr and stu are similar by sas similarity

Explanation:

Step1: Recall SAS similarity criterion

SAS (Side - Angle - Side) similarity criterion states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, then the two triangles are similar.

Step2: Analyze each option for the first question

• Option a: Verifying similarity between two mountain slopes can use SAS similarity (if two sides and included angle are considered).
• Option b: Proving congruence of two triangles uses SAS congruence (not similarity, but the question is about not using SAS similarity. Congruence is a different concept but the key is the nature of the operation. SAS congruence is not SAS similarity).
• Option c: Determining similarity of two bird nests based on size ratios (if we assume two sides and included angle, it can be SAS similarity).
• Option d: Proving that two bridges have proportional supports (if we consider two sides and included angle, it can be related to SAS similarity).

Step3: Analyze the second question

Given \(\angle P=\angle S\) and \(\frac{PQ}{ST}=\frac{PR}{SU}\). By the SAS (Side - Angle - Side) similarity criterion (since the angle is the included angle between the two pairs of proportional sides), triangles \(PQR\) and \(STU\) are similar by SAS similarity.

Answer:

For the first question: B. Proving congruence of two triangles
For the second question: D. Triangles \(PQR\) and \(STU\) are similar by SAS similarity